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Sylow \(p\)-subgroups of Abelian group rings. (English) Zbl 1035.16025

Let \(KG\) be the group ring of an Abelian group \(G\) over a commutative ring \(K\) and let \(V(KG)\) be the group of normalized units (i.e. the group of augmentation 1) in \(KG\). In the paper some group-theoretic properties of the \(p\)-component \(S(KG)\) of \(V(KG)\) are proved when either (i) \(K\) is a ring of prime characteristic \(p\) or (ii) \(K\) is a field of the first kind with respect to \(p\) and the characteristic of \(K\) is different from \(p\). The author considers case (ii) under the following restriction: the spectrum of \(K\) with respect to \(p\) contains all naturals. The considered properties are very general and do not give the structure of \(S(KG)\).
The most interesting statement is Theorem 7 for the semisimple case, i.e. for case (ii). It states that if \(A\) is a direct sum of cyclic \(p\)-groups then \(A\) is a direct factor of \(S(KA)\), and consequently of \(V(KA)\). But a full description of \(S(KA)\), to within isomorphism, is given by the reviewer [PLISKA, Stud. Math. Bulg. 8, 34-46 (1986; Zbl 0662.16008) and ibid. 8, 54-64 (1986; Zbl 0655.16004)]. Therefore Theorem 7 does not have a big value. Besides, the proof is incomplete, since it is based on Lemma 6, in which it is not proved that the second factor in the decomposition of \(S(RG)\) is a subgroup of \(S(RG)\). We note that this fact is obvious only in the modular case.
The second proof of Proposition 1 is not true. We can give the following counterexample to this proof. Let \(A\) be the direct product of the groups \(H\) and \(R\), where \(H\) is an unbounded direct sum of cyclic \(p\)-groups and \(R\) is a cyclic group of order \(p\). Then \(A^p=H\) holds, i.e. \(t=1\). We choose \(H_k=H\) for every natural \(k\). Then the indicated cross-section on p. 35, line 6(–), is not 1, which contradicts the author’s statement.
The proofs of properties (4) and (5) in Theorem 4 are not true, since the decompositions, which are indicated, are valid only when the group \(A\) is infinite. Theorem 5 is superfluous, since it can be trivially obtained from Theorem 4.
The author discusses on page 45 a result of Nachev’s and at the same time he is citing his own paper. This is incorrect.
The paper is not written clearly. For example, in the cited Theorem [4] at the beginning of section 2, he does not define the group \(V(PG;H)\), although he preliminarily writes “for the convenience of the reader …”. At the same time it is not understandable how Theorem [4] is used in the proof of Proposition 2. In the proof of Theorem 7 the author uses, without citing, that \(S(KH_n)\) has finite exponent \(n\), which is the reviewer’s result (see the above cited papers) and it is not obvious.

MSC:

16U60 Units, groups of units (associative rings and algebras)
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16S34 Group rings
20E07 Subgroup theorems; subgroup growth
20K21 Mixed groups
20K25 Direct sums, direct products, etc. for abelian groups
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